Mathematical analysis and Lebesgue integral

• Grant number: 11/19146-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: November 01, 2011
• End date: October 31, 2012
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Marcio Fuzeto Gameiro
• Grantee: Diego Sousa Garcia Sanz
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil

Abstract

The goal of this project is the study of analysis, metric spaces and Lebesgue integral, in order to deepen the knowledge mathematics of students and give them a solid foundation for better development their academic and professional lives. This project is divided into four parts. The first is for the study of metric spaces and some basic properties such as compactness and connectedness. The second part is intended for studies of numerical sequences and series, continuity of functions and the differentiability of real valued functions. The third part of this project is focused on studies of sequences and series real valued functions, with the main results of The Arzel-Ascoli and Stone-Weiertrass Theorems. The fourth and final part is devoted to the study of measure theory, specifically the Lebesgue integral and its applications, such as integrals of functions dependent on parameters and derivation under the integral sign. For this purpose, the books will be studied "Principles of Mathematical Analysis" by Walter Rudin and "The Lebesgue Integral and its Applications " C. S. Honig. The student will use the AMS-LATEX tool for typing reports. (AU)